Chiral corrections to $\pi^-\gamma\to 3\pi$ processes at low energies

TL;DR

This paper calculates low-energy chiral corrections to pion photoproduction processes, providing analytical amplitudes and cross section predictions that can be tested in upcoming experiments, enhancing understanding of pion interactions.

Contribution

It presents the first detailed next-to-leading order chiral perturbation theory calculations for $\pi^-\gamma o 3\pi$ processes, including analytical amplitudes and invariant mass spectra.

Findings

Next-to-leading order corrections enhance neutral pion-pair production cross section by 1.5-1.8 times.

Charged pion-pair production cross section remains nearly unchanged at NLO.

Predictions are testable by the COMPASS experiment.

Abstract

We calculate in chiral perturbation theory the double-pion photoproduction processes $\pi^-\gamma \to \pi^- \pi^0\pi^0$ and $\pi^-\gamma\to \pi^+\pi^-\pi^-$ at low energies. At leading order these reactions are governed by the chiral pion-pion interaction. The next-to-leading order corrections arise from pion-loop diagrams and chiral-invariant counterterms involving the low-energy constants $\bar\ell_1,\, \bar\ell_2,\, \bar\ell_3$ and $\bar\ell_4$. The pertinent production amplitudes $A_1$ and $A_2$ depending on five kinematical variables are given in closed analytical form. We find that the total cross section for neutral pion-pair production $\pi^-\gamma \to\pi^-\pi^0\pi^0$ gets enhanced in the region $\sqrt{s}< 7m_\pi$ by a factor $1.5 - 1.8$ by the next-to-leading order corrections. In contrast to this behavior the total cross section for charged pion-pair production $\pi^-\gamma\to \pi^+ \pi^-\pi^-$ remains almost unchanged in the region $\sqrt{s}< 6m_\pi$ in comparison to its tree-level result. Although the dynamics of the pion-pair production reactions is much richer, this observed pattern can be understood from the different influence of the chiral corrections on the pion-pion final state interaction ($\pi^+\pi^- \to \pi^0\pi^0$ versus $\pi^-\pi^- \to \pi^-\pi^-$). We present also results for the complete set of two-pion invariant mass spectra. The predictions of chiral perturbation theory for the $\pi^-\gamma\to 3\pi$ processes can be tested by the COMPASS experiment which uses Primakoff scattering of high-energy pions in the Coulomb field of a heavy nucleus to extract cross sections for $\pi^-\gamma$ reactions with various final states.